Let us consider the propagation of a linearly polarised beam of coherent light through the
nonlinear optical medium of a cell filled with a nematic liquid crystal [1]}, [2]}. The
light is assumed to propagate in the direction \(z\)  down the cell with the beam polarised in
the \(y\)  direction. The \(x\)  coordinate then completes the coordinate triad. Typical physical
cell dimensions are \(200\mu \mbox{m}\times 30\mu \mbox{m}\times 1\mbox{mm}\)  [1]}. Nematic molecules are biaxial [1]}
with a tensor refractive index. There are then two eigendirections for optical beams, termed the ordinary and extraordinary
polarisations [5]}. Only the extarordinary polarisation is dispersive and able to support solitary waves
[5]}. Hence, the optical beam will be assumed to be extraodinarily polarised. An added complication
of the nematic medium is that if the nematic molecules are initially aligned with
their axis, termed the director, orthogonal to the electric field, the optical
Freédericksz threshold exists so that a minimum electric field strength is required
to overcome the elastic forces of the nematic medium before the molecules can rotate
[7]}, [1]}. To overcome this threshold so that milliwatt power optical beams can rotate the nematic
molecules (and so change the refractive index of the medium), an external static electric field is applied to
pre-tilt the nematic molecules at an angle \(\theta _{0}\)  [9]}. Low power optical beams are necessary
so that there is minimal heating of the nematic as excessive heating can cause the medium to change phase out of
the nematic state. A nematic is typically a focusing medium, so that rotation of the nematic molecules by an
optical beam increases the refractive index, leading to self-focusing [1]}, [2]}. However, the formation of
a DSW requires the medium to be defocusing [12]}. It has been found that the addition of small amounts
of azo dye to the nematic changes the medium response, so that the nematic becomes defocusing [13]}. The
addition of the dye increases scattering losses for the optical beam, but these will be neglected in the present work.
With this background and these assumptions, in the paraxial, slowly varying envelope approximation [5]},
the non-dimensional equations governing the propagation of an optical beam in a defocusing nematic are
[1]}, [2]}, [13]}, [18]}
\(i \frac{\partial u}{\partial z} + \frac{1}{2}\frac{\partial ^{2} u}{\partial x^{2}} - 2\theta u & = & 0 , \\\nu \frac{\partial ^{2}\theta }{\partial x^{2}} - 2q\theta & = & - 2|u|^{2} . \) 
